
theorem Th41:
  for N being invertible Matrix of 3,F_Real
  for p,q,r,s being Element of ProjectiveSpace TOP-REAL 3 st
  Line(homography(N).p,homography(N).q) = Line(homography(N).r,homography(N).s)
  holds p,q,r are_collinear & p,q,s are_collinear &
        r,s,p are_collinear & r,s,q are_collinear
  proof
    let N being invertible Matrix of 3,F_Real;
    let p,q,r,s being Element of ProjectiveSpace TOP-REAL 3;
    assume
A1: Line(homography(N).p,homography(N).q)
      = Line(homography(N).r,homography(N).s);
    hence p,q,r are_collinear by ANPROJ_8:102,Th39;
    Line(homography(N).p,homography(N).q)
      = Line(homography(N).s,homography(N).r) by A1,Th40;
    hence p,q,s are_collinear by ANPROJ_8:102,Th39;
    thus r,s,p are_collinear by A1,ANPROJ_8:102,Th39;
    Line(homography(N).q,homography(N).p)
      = Line(homography(N).r,homography(N).s) by A1,Th40;
    hence r,s,q are_collinear by ANPROJ_8:102,Th39;
  end;
